OF THE MOVEMENT OF WORMS
BY GARTH CHAPMAN
From the Department of Zoology, Queen Mary College, University of London
(Received 10 June 1949)
(With Three Text-figures)
CONTENTS
PAGE
I.
II.
III.
IV.
V.
VI.
Introduction
.
.
.
.
.
.
.
.
.
.
T h e antagonism of circular a n d longitudinal muscles
.
Hydrostatic pressure and muscular tension
.
.
.
.
T h e application of the internal pressure t o the substratum
Changes of muscle length and body shape
.
.
.
.
Summary
.
.
.
.
.
.
.
.
.
.
References
.
.
.
.
.
.
.
.
.
I.
.
.
29
30
33
36
37
39
39
INTRODUCTION
There is little published work dealing with the movement of worm-like animals,
apart from descriptive papers such as that of Foxon (1936), and analyses of the
application of muscular contraction to the substratum such as are found in the
papers of Gray (1939) and Gray & Lissmann (1938a, b). The possession of the right
volume of coelomic fluid was shown by Chapman & Newell (1947) to be important
in the burrowing of the lugworm, and the function of the body fluid in the extrusion
of the proboscis of Nephthys and in the movements of Calliactis has been demonstrated (Chapman, 1949 and 1950). There appears, however, to be no satisfactory
account of the interaction of the antagonistic muscles which is made possible by the
presence of the bodyfluid,although von Buddenbrock (1937) mentions the antagonism
between the longitudinal and circular muscles and describes their nervous coordination.
In the present paper an attempt is made to examine theoretically the working of
antagonistic longitudinal and circular muscles in cylindrical fluid-filled animals.
In the movement of these creatures there are features upon which it is necessary to
lay stress, and although they may not all be self-evident they may, for convenience,
be stated concisely until a more rigorous proof can be given.
(a) The longitudinal and circular muscles are antagonists which depend for their
antagonism upon the presence within the animal of a fluid-filled cavity of fixed
volume. (The fluid-filled cavity varies in size in different animals and although, in
many, the volume of coelomic fluid is small, nevertheless that which is contained
within the body wall may be regarded as fluid and incompressible for the purposes
under review.)
(b) The maximum thrust which the musculature of an animal can exert is measured
by the internal hydrostatic pressure and the area of application. The maximum
30
GARTH CHAPMAN
pressure can be increased above the internal hydrostatic pressure by the application
of the principle of the wedge.
(c) The maximum thrust which the animal can exert is commensurate with the
friction forces available to-resist it.
(d) The absolute change in length of the animal which can be brought about by
a given percentage contraction of the circular muscles depends on the shape of the
animal, i.e. upon the relation of length to diameter.
It is necessary, then, to put forward some justification for the previous statements,
and to examine their bearing on the shapes, sizes and muscular construction of
soft-bodied invertebrates.
II. THE ANTAGONISM OF CIRCULAR AND LONGITUDINAL MUSCLES
It has been understood since the time of Leonardo da Vinci that a muscle ' uses its
power along the line of its length'. In general terms this power is exerted during
contraction, and a muscle is unable to exert its power again until it is restored to its
resting length by an external agent. Spontaneous elongation of the muscles of the
starfish podium has been reported by Paine (1929), but Smith (1946) has clearly
refuted the suggestion and has collected considerable experimental and histological
evidence to show that it does not occur, the elongation of the isolated tube foot being
due to the unfolding of the crumpled collagen fibres which can exert sufficient force
slowly to stretch the muscle.
To restore its 'useful' (relaxed) state a muscle may have in opposition to it either
the force of another muscle, an elastic force or possibly a ciliary pressure. It may be
noted that the force developed by the restoration system need not be as great as that
developed by the effector. If the muscle works against an elastic restoring agent
the work which it can perform externally is diminished by the work which is required
to deform the elastic body.
The deformation of an elastic substance and the use of ciliary pressure cannot be
regarded as important methods of restoration in the animal kingdom as a whole,
although elasticity may play a part in the functioning of the muscular system of
nematodes and ciliary pressure in that of coelenterates. Whilst it is true that ciliary
pressures are of the order of mm. of water only (Pantin, private communication),
it is also true that very small forces suffice during life to extend the body wall of
certain actinians (see, for example, Jordan, 1935)If a muscle works against an antagonistic muscle the work which it can perform
externally is diminished only by the work necessary passively to extend the antagonist
through the transmission system by which its force is applied. In many of the lower
invertebrates this takes the form of a hydrostatic skeleton consisting of afixedvolume
of liquid or tissue which is in a practically fluid state during life. The liquid may be
the contents of the enteron, the coelom or the haemocoel, but one of its functions in
all instances is that of a pressure-transmission system which causes the effects of
contraction of any one muscle automatically to be felt by all the remaining muscles
in the body wall. The contraction of a muscle in the body wall running parallel to
the surface would cause a diminution in the volume enclosed by the body wall if
Of the movement of worms
31
the fluid were compressible. Since, however, it is incompressible the contraction of
any one muscle must bring about an increased internal pressure and an extension of
the remaining muscles.
Any fluid-muscle system which consists of a fluid surrounded by a muscular wall
has inherently in its construction its muscular restoring agent, provided that muscle
fibres of the wall run in longitudinal and circular directions, or that the skeletal
materials of the wall contain an element of elasticity in their physical properties.
This can be seen by considering a hypothetical animal whose body wall consists of
circular fibres only and lacks any skeletal elasticity. Contraction of the muscles at
one end could bring about a thinning of that end, and either an elongation or
a thickening at the other (see Fig. 1). If this thickening were to take place, all that
•(; 0 0 0 0 U L X i U
Body elongating
Circulars contracting
0 (rrro
Body thickening
Circulars contracting
Fig. 1. Diagram to illustrate the action of a hypothetical animal with circular muscles only. In A all
the muscles are relaxed. In B the muscles of the right-hand end have contracted but the length
of this end has remained the same. If this takes place it is impossible for the original length of
the right-hand muscles to be restored by the contraction of the left-hand muscles. In C an
alternative arrangement is shown. The muscles of the right-hand end have contracted, but the
muscles of the left-hand end have relaxed. There hag been no change in length. By the contraction
of the left-hand muscles the right-hand muscles could be restored to their original length.
would be necessary to restore the original condition would be a contraction of the
dilated end. But, on the other hand, if an elongation were to occur, there would be
no way of getting back to the original condition since contraction of the muscles in
the elongated region would serve only to relax the muscles in the original contracting
end, and so to thicken it without shortening the body as a whole (see Fig. 1). It is
interesting to note here that, where only a single muscular coat occurs in a hollow
organ 9uch as a starfish podium, there is a strong inelastic sheath of collagen fibres
which prevents increase in diameter of the foot on contraction of its muscles. When
this occurs the contained fluid is driven into the ampulla, which together with the
tube foot, acts as a closed system.
In most soft-bodied invertebrates the musculature of which the body wall is
composed is arranged according to a uniform plan in which the outer layer consists
of fibres running in a circular manner, whilst the inner layer of fibres runs along the
32
GARTH
CHAPMAN
length of the animal. This arrangement enables the muscles to act antagonistically to
one another without the necessity for the body wall to possess any skeletal elastic
component, provided that the nervous system arranges for the muscles to be
suitably stimulated. If the action of an animal in which there are circular and
longitudinal muscles be compared with that of the hypothetical organism consisting
of circular muscles only, it can be seen that the contraction of the circular muscles
at one end can bring about four different muscular configurations which are
A A A A A
\
A
p
i
i
<
>
•
V
u
i
1
1
V
V
T
T
7i
T
\: I (:
*,
/\
i
/
i
i
7
if
"/.' ' A "~~f " T '
'
U
( '
V
1
L
/
1
V
I
Fig. 3. Diagram to illustrate four possible results of the contraction of circular muscles at one end
of a cylindrical animal. In A the muscles are all relaxed. In B the circular muscles of the
right-hand end have contracted and this end has elongated. The left-hand end has remained
unaltered. In C the length of the right-hand end has remained the same but the diameter of the
left-hand end has increased. In D the length of the right-hand end has also remained the same.
The length of the left-hand end has increased but not its diameter. In £ the length of both
ends has increased but their diameters have remained the same as in B and D.
illustrated in Fig. 2. Restoration of the original body shape from any of these states
can be brought about by the contraction of appropriate other parts of the musculature
as can be seen on inspection of the figure.
That the ordering of the muscle fibres in a longitudinal and circular manner is the
only arrangement which makes possible the movements of soft-bodied invertebrates
can be seen by considering the results of the arrangement of the fibres at an angle of
450 to the long axis in such a way that they are arranged in two spirals of opposite
sense. The effect of the contraction of one spiral on the dimensions of the body
Of the movement of worms
33
would be the same as that of the other so that neither could be said to be in opposition
to the other nor do they constitute a mutually restorative system. On the other
hand, the arrangement of inextensible skeletal fibres in this manner allows change of
dimensions of the animal at fixed volume without stretching of the skeletal material.
The antagonism of relaxed muscles to those which are contracting depends on
the presence of a body fluid which in its turn necessitates a non-leaking body wall.
In those animals which have been examined this has been found to occur. In
Arenicola the nephridiopores are sphinctered apertures capable of resisting a pressure
greatly in excess of that normally produced by contraction of the body-wall muscles.
Loss of body fluid causes loss of burrowing efficiency, which is prevented on mutilation by the marked constriction of the body wall on to the gut (Chapman & Newell,
1947). In CalUactis practically no water leaves the stomodaeum or the cinclides on
normal contraction (Chapman, 1950). In Lumbriais free flow of fluid about the
body is prevented, and escape of fluid on mutilation is hindered, by the presence of
muscular septa; neither do the dorsal pores nor the nephridiopores leak during life
(Newell, private communication).
III. HYDROSTATIC PRESSURE AND MUSCULAR TENSION
The relation between the pressure in a hollow cylinder and the stresses in its walls
can be seen by considering the forces exerted on a longitudinal section passing
through the axis and also the forces on a cross-section.
If 't' is the thickness of the wall, ' / ' t h e length, ' r ' the radius and '/<,' and'/,' the
circumferential and longitudinal stresses respectively, the total push due to the
pressure on one half of the longitudinal section equals pzrl. This must be balanced
by the circumferential stress in the walls, viz. fpXl. Therefore
Similarly, for the cross-section the total pressure on the ends must be balanced by
the longitudinal stresses in the walls, viz. pnrl = iiTrtfl. Therefore
ZTTTtfj
fj2t
Considering now the tensions in the longitudinal and circular muscles of a worm,
and the pressures which they exert on the contained fluid, Tc can be written instead
of the expression fjl for the circumferential stress as being the total tension in the
whole circular muscle layer so that
pzrl=2Te,
iT
T
Therefore
p = :t±j = ±f.
r
zrl rl
Similarly, the total tension in the whole longitudinal muscle layer, Tt, can be
written for the expression 2Trrtfj so that the equation becomes
Therefore
P= ^ \
r
J KB.27, I
77T2
GARTH CHAPMAN
34
From the expressions relating pressure with longitudinal and circumferential
stresses it can be seen that
1 C
1 j
1 C
1 i
1 i
TTT
1TT
It follows from this expression that if the ratio of the tensions in the longitudinal
and circular muscles is equal to irrjl, then the pressures being exerted on the contained fluid by the two muscle layers are equal. Whilst, therefore, there is no doubt
that this equation may hold good at any instant, it is of interest to inquire if the
quantities of longitudinal and circular muscle present in a typical 'worm' are such
that the equation holds good when both layers are contracting maximally. This can
be discovered by inspection, if it is assumed that the tension which a muscle can
exert is proportional to its cross-sectional area. If Ax and Ac are the total crosssectional areas of the longitudinal and circular muscles respectively, then
A1
TTT
Average measurements made from camera Uicida drawings of sections of earthworms of different sizes are set out in Table i.
Table i. Dimensions of body-wall muscles of Lumbricus
(Dimensions in mm. and sq.mm.)
Large
worm
Approximate length before fixation
Radius
Radial thickness of circular muscles
Total cross-sectional area of circular muscles
Radial thickness of longitudinal muscles
Total cross-sectional area of longitudinal muscles
AJA0 (approx.)
w/i
150
Medium
worm
no
Small
worm
60
27
i-8
I'O
O'l
0-05
5'5
c-3
0-04
15-0
0-3
5'9
0-4
0-05
3-0
0'5
C05
2-4
0'2
1-2
O'5
0-05
It can be seen from the table that in the earthworm Aj/Ac is not equal to -rrr/l,
implying either that the tension exerted by the muscles is not proportional to their
cross-sectional area or, which is much more likely, that at their maximal contraction
the circular and longitudinal muscles are not balanced. From the values of TTT/1
calculated from the values of 'r' and ' / ' i t would appear that, for equal pressure to
be exerted by circular and longitudinal layers, Tc should be approximately 20 x Tt
(or Ac should be approximately equal to 2OxAj) whereas, in fact, Ae is only
approximately 2 x A{. If, then, both sets of muscles in the worm were to contract
maximally, the pressure exerted by the longitudinals (TJ/TTT2) would be approximately ten times that exerted by the circulars (TJrT).
Since contraction of the circular muscles gives rise to a forward thrust they might
have been expected to be the stronger, for they would seem to be the more important
both for progression and for burrowing through the soil. However, since the animal
probably burrows largely through existing crevices, the longitudinal muscles may be
of greater importance since they serve to increase the diameter of the body and
Of the movement of worms
35
hence to thrust aside the soil particles. (This argument presupposes, of course, that the
body is totally enclosed by the soil, because otherwise the longitudinal muscles can
generate only as much pressure as the circular muscles are capable of withstanding.)
The maximum pressures in both Aremcola and Lumbricus occur during phases of
contraction of the anterior circular muscles when the animals are unconfined, but
in Aremcola, enclosed in a glass tube, values of between 80 and 90 cm. of sea water
have been recorded as compared with the highest values of 50-60 cm. of sea water
previously registered (Chapman & Newell, 1947). This pressure is not as great as
the theoretical maximum but it shows, nevertheless, that greater pressures can be
exerted by the longitudinal than by the circular muscles.
It is clear that the hydrostatic pressure in the bodyfluidis a measure of the pressure
which can be applied to the substratum, so that the measurement of hydrostatic
pressure is a convenient way of measuring the' strength' of soft-bodied invertebrates,
although the total pull which can be exerted by an animal during the contraction of
its longitudinal muscles cannot be measured by the measurement of hydrostatic
pressure since the transmission of the muscular tension does not require the intervention of any fluid. In this connexion it is interesting to note that the figure given
by Gray & Lissmann (1938a) for the maximum pull which an earthworm can exert
(70 g.) is in excess of the thrust which it normally exerts on extension (2-8 g.).
This figure of 2-8 g. is in good agreement with the hydrostatic pressure as measured
by Newell. His values have a maximum of about 30 cm. of water, i.e. 30 g./sq.cm.
For a worm of diameter o-6 cm., of cross-sectional area 0-28 sq.cm., the total thrust
would therefore be about 8-5 g.
In order to compare the pressure exerted by the circular muscles with the hydrostatic pressure it is necessary to assume, as has been done before, that the tension
which a muscle can exert is proportional to its cross-sectional area. If this is done,
and the pull exerted by the longitudinal muscles is used as a standard, it is possible to
compute the pressure which ought to be exerted by the contraction of the circular
muscles. If the pull due to the longitudinal muscles of an earthworm of about
0-3 cm. radius is 70 g., the cross-sectional area of its longitudinal muscles being
0-06 sq.cm., then the tension in the circular muscles, of cross-sectional area
0-15 sq.cm., should be
j-=- g. The pressure exerted on the contained fluid
O"OO
(TJrt) should therefore be —-.
v
"' '
— g./sq.cm. This value, of 38 g./sq.cm., is
0-06x0-3 x 1 5 6 '
M
>
J 6/ M
.
again in quite good agreement with the measured value of 30 cm. of water.
It is interesting to notice here that the pressure exerted by the circular muscles is
not dependent upon their simultaneous contraction along the total length of the
animal if the creature is capable of being divided into water-tight compartments.
If, for example, a third of the length is in contraction, then the cross-sectional area
of the circular muscles in this portion will be a third of the total circular muscles in
the body, but so will the value of '/' in the equation
T
pressure = —3-2
36
GARTH CHAPMAN
If, on the other hand, the body cavity is open and the fluid in the region undergoing contraction is in communication with the remainder, then the development
of pressure in the body cavity is dependent upon the development of tension in
the muscles as a whole. Examples of the open and septate coelom are seen in the
lugworm and the earthworm respectively.
IV. THE APPLICATION OF THE INTERNAL PRESSURE
TO THE SUBSTRATUM
Whether the thrust is applied by the extrusion of the proboscis as in Nephthys and
Nereis, or by the extension of the anterior end as in Lumbricus, the effective application of the thrust resulting from the contraction of the body-wall muscles depends
upon the ability of the animal to withstand the reaction to the thrust.
Although it may perhaps unduly extend the meaning of the term friction to include
the resistance provided by parapodia and setae, nevertheless the forces exerted by
these portions of the animal must be included with the friction of the body. Frictional
forces have been measured for the earthworm by Gray & Lissmann (1938 a), but no
measurements of the frictional forces of the animals in their natural environment
appear to have been made. Friction between an earthworm and the walls of its
burrow may be considerable, since animals can easily be broken when attempts are
made to pull them out. Presumably, therefore, if the longitudinal muscles of
Lumbricus are capable of exerting a tension of 70 g. the frictional resistance which
must be overcome in pulling a worm from its burrow can be at least 70 g.
It is also worth comment that the contraction of the longitudinal muscles serves to
increase the diameter of the body and therefore may increase the pressure applied at
right angles to the surface of the body of the worm.
If the frictional force which must be overcome for movement to occur be '/',
then the coefficient of friction is//n, where 'n' is the normal reaction between the
applied surfaces. In Gray & Lissmann's example of an earthworm moving over
a glass plate the frictional resistance was 2-8 g., say 5 g., and the weight of the worm
was, say, 5 g. The coefficient of friction is therefore 1. If the coefficient of friction
is the same for a worm being pulled from its burrow and if f=jo g., then, since
70/71=1, 71 = 70, or the total reaction between the worm and its burrow is 70 g.
Taking the superficial area of the worm to be 25 sq.cm. the pressure of the worm on
its burrow is 70/25 g./sq.cm. This figure, of less than 3 g./sq.cm., is obviously well
within the limits of the pressure occurring in the body fluid. Even if it were to be
multiplied by a factor of 10, as would be necessary if only one-tenth of the superficial area of the worm were in contact with the wall of the burrow, it would imply
a pressure of 30 g./sq.cm., which it is still well within the animal's power to generate.
It would appear to be fallacious to assume, as has in fact been done, that the
coefficient of friction between the earthworm and a smooth plate is the same as thaf
between a worm and its burrow, although the presence of mucus in both instances
might well have the effect of lessening any differences. Some rough measurements
with a dynamometer, made by dragging eight fresh Arenicola along the surface of
drained sand, showed that each animal required a force of 10 g. weight to move it
Of the movement of worms
37
against the force of friction. The freshly dug worms which were used in this experiment weighed about 10 g. each so that the coefficient of friction between drained
sand and the animal was about 1, which does not differ from the value deduced from
Gray & Lissmann's observations on the earthworm. Forces involving the displacement of particles of muddy sand are not reliable quantities to measure, and not too
much significance can be attached to these results.
To sum up, the figure of 30 g./sq.cm. of water necessary for an earthworm to
resist a pull of 70 g. when one-tenth of its length is in its burrow is well within the
maximum pressure which the worm is capable of exerting.
Newell (private communication) has pointed out that an earthworm can increase
the pressure which it applies to the substratum by reducing the area in contact with
the soil. If the anterior end of the animal is regarded as a truncated cone and if the
internal pressure be applied to the larger end of the cone, the pressure exerted on
the soil by the small end will be in the inverse ratio of the areas of the two ends. In
this way, working at an internal pressure of 30 g./sq.cm., and with the area of
application of the force one-twentieth that of the cross-sectional area of the worm,
a force of 600 g./sq.cm. or 8 lb./sq.in. can be applied. If the total cross-sectional
area of the worm is 0-25 sq.cm., then the diameter of the smaller area of application
of the force would be between 1 and 2 mm. This figure certainly appears to be of the
right order. It will be realized that this is a considerable pressure when compared
with that of 2-5 lb./sq.in. exerted on the ground by a man standing on both feet;
and it is highly likely that the tissue of the animal would be deformed by such
pressure so that the area of contact would be immediately increased. It is worth
noting, however, that the anterior end of an earthworm is a nearly solid muscular
region which can be made rigid by contraction. It will also be seen that, since the
forward thrust is expressed as a pressure, i.e. force per unit area, it implies that no
increase is necessary in the frictional forces (between the animal and the ground)
which resist the forward thrust.
V. CHANGES OF MUSCLE LENGTH AND BODY SHAPE
In their simplest form, taking no account of the changes in shape of the muscles
themselves, the variation in length and circumference of a cylindrical animal can
be regarded as those changes which take place in a cylinder of variable dimensions
and fixed volume, V.
V=r^l=z constant.
Therefore
1 = —2,
from which it can be shown that
dl
I
-zdr
(Blaney, private communication).
That is, a percentage change in length is accompanied by half that percentage change
in circumference. For example, the dimensional changes in a cylinder of fixed
38
GARTH CHAPMAN
volume of 15,700 cubic units are given in Table 2, a radius of 5 units and a length
of 200 units being proportions not unlike those of a 'worm*.
Table 2. Variations in the length, circumference and radius
of a cylinder of fixed volume
(Volume (F) = 15,700 cubic unita, r = radius, c = circumference, Z=length.)
T
c
/
V
2
12-5
1260-0
3
189
SSS-6
4
5
6
25-1
31*4
aoo-o
377
43'9
138-9
IO2-I
15,700
15,700
15.700
15.700
15,700
15,700
15,700
I
312-4
78-O
5O-3
The graph which can be drawn from the relation between ' / ' and 'c' is shown in
Fig. 3, and shows clearly that for a given percentage change in length of the circular
100 20
30
Circumference
Fig. 3. Graph showing the relationship between the length and circumference
of a cylinder of fixed volume.
muscles the longitudinals change in length by twice that percentage. Alternatively,
for a given absolute change in length of the circular muscles the change in length
of the longitudinals produced by it may be great or small according to whether
l/r is at its greatest or smallest. An implication of this is that, since the smallest
change in length of the animal for a fixed change in length of the circulars occurs
when the animal is at its 'fattest', it would be expected that burrowing animals
would carry out their burrowing when thick and not when thin. Conversely, when
Of the movement of worms
39
an animal is crawling it will, for each fixed contraction of the circular muscles,
progress farthest if it is thin. If, for example, the working range of an earthworm
were 25 % of its resting length, then, although this means a 50% change in length
for the longitudinals, it involves only a normal working range of 25 % for the circulars.
Whether this difference in requirements is based on any difference in the physiology
of the muscles is unknown.
VI. SUMMARY
Four aspects of the functioning of a fluid-filled cylindrical animal have been
examined, viz.: (1) the role of the body fluid as a skeleton for the interaction of the
longitudinal and circular muscles of which the animal must be composed; (2) the
measurement of the maximum thrust which the animal can exert by measurement
of its internal hydrostatic pressure; (3) the application of the force to the substratum
and the part played by friction; (4) the relation between the changes in dimensions
of the animal and the working length of the muscles.
Under (1) the necessity for a longitudinal and circular construction has been
shown and the necessity for a closed system emphasized.
Under (2) the pressure exerted on the body fluid by the contraction of the longitudinal and circular muscles is discussed, and from their cross-sectional areas it is
shown to be probable that when contracting maximally in Lumbricus they are not
balanced, but that the longitudinals are about ten times as strong as the circulars.
Under (3) it is shown that the strength of an animal as measured by its internal
hydrostatic pressure is sufficient to account for its customary activities. Use which
may be made of the longitudinals during burrowing is pointed out.
. Under (4) it is shown to be mechanically sound for burrowing animals of cylindrical
form to be 'fat', but that a 'thin' animal is more efficient at progression.
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GRAY, J. & LISSMANN, H. W., (19386). J. Exp. Biol. 15, 518.
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NBWELL, G. E. Private communication.
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PANTIN, C. F. A. Private communication.
SMITH, J. E. (1946). Pkilos. Trans. B, 333, 279.